Existence and asymptotic behavior of solitary waves for a weakly coupled Schrödinger system

Abstract

Abstract This paper deals with the following weakly coupled nonlinear Schrödinger system − Δ u 1 + a 1 ( x ) u 1 = ∣ u 1 ∣ 2 p − 2 u 1 + b ∣ u 1 ∣ p − 2 ∣ u 2 ∣ p u 1 , x ∈ R N , − Δ u 2 + a 2 ( x ) u 2 = ∣ u 2 ∣ 2 p − 2 u 2 + b ∣ u 2 ∣ p − 2 ∣ u 1 ∣ p u 2 , x ∈ R N , \left\{\begin{array}{ll}-\Delta {u}_{1}+{a}_{1}\left(x){u}_{1}=| {u}_{1}{| }^{2p-2}{u}_{1}+b| {u}_{1}{| }^{p-2}| {u}_{2}{| }^{p}{u}_{1},& x\in {{\mathbb{R}}}^{N},\\ -\Delta {u}_{2}+{a}_{2}\left(x){u}_{2}=| {u}_{2}{| }^{2p-2}{u}_{2}+b| {u}_{2}{| }^{p-2}| {u}_{1}{| }^{p}{u}_{2},& x\in {{\mathbb{R}}}^{N},\end{array}\right. where N ≥ 1 N\ge 1 , b ∈ R b\in {\mathbb{R}} is a coupling constant, 2 p ∈ ( 2 , 2 ∗ ) 2p\in \left(2,{2}^{\ast }) , 2 ∗ = 2 N / ( N − 2 ) {2}^{\ast }=2N\hspace{0.1em}\text{/}\hspace{0.1em}\left(N-2) if N ≥ 3 N\ge 3 and + ∞ +\infty if N = 1 , 2 N=1,2 , a 1 ( x ) {a}_{1}\left(x) and a 2 ( x ) {a}_{2}\left(x) are two positive functions. Assuming that a i ( x ) ( i = 1 , 2 ) {a}_{i}\left(x)\hspace{0.33em}\left(i=1,2) satisfies some suitable conditions, by constructing creatively two types of two-dimensional mountain-pass geometries, we obtain a positive synchronized solution for ∣ b ∣ > 0 | b| \gt 0 small and a positive segregated solution for b < 0 b\lt 0 , respectively. We also show that when 1 < p < min { 2 , 2 ∗ / 2 } 1\lt p\lt {\rm{\min }}\left\{2,{2}^{\ast }\hspace{0.1em}\text{/}\hspace{0.1em}2\right\} , the positive solutions are not unique if b > 0 b\gt 0 is small. The asymptotic behavior of the solutions when b → 0 b\to 0 and b → − ∞ b\to -\infty is also studied.